Bartlett-type corrections for the confidence interval of a treatmenteﬀect of the multivariate random eﬀects meta-analysis model via
analytical approach
Masahiro Kojima, Biometrics Department, R&D Division, Kyowa Hakko Kirin Co., Ltd.
Over the past few decades, many researchers have shown an interest in meta-analysis which has been
active in various fields: ecology, economics, medical science, medicine, and so on. As Gurevitch et al. (2018)indicates, the research with meta-analysis began to be published in 1970s, its number rapidly has increasedin recent years, and has exceeded 200,000 papers. As far as the clinical trial is concerned, the number ofstudy has risen to start to be conducted in some countries year by year, the research with meta-analysis
seems likely to expand further in the future.
Meta-analysis is to synthesize the measured eﬀect of treatments or correlated outcomes (ex. overall anddisease-free survival) of high quality studies via the systematic reviews. It should be noted that there isheterogeneity between studies, assuming that each study has true results, respectively. In order to address
this problem, we consider the random eﬀect model. The random eﬀect has the advantage of expressing thediﬀerence between studies as its distribution. Thus, this research involves the multivariate random eﬀects
meta-analysis model.
In the context of meta-analysis, the treatment eﬀect size of each study have been already known, andaverage treatment eﬀect size of interest is estimated by summarizing studies. Therefore, the accuracy ofoverall mean estimator depends the number of study. As Noma et al. (2018) points out, the number of study
is small in the case conducted subgroup analysis, the study of the orphan disease is few to rare disease[3]. So
as to more increase the accuracy in this situation, Noma et al. (2018) proposed the bartlett correction via
bootstrap. However, the problem seems to lie in the two facts: (1) bootstrap calculation takes much time tonumerically solve maximum likelihood (ML) estimator or restricted maximum likelihood (REML) estimator,(2) the confidence interval cannot be expressed in closed form, and has to be derived by numerical approach,it also takes time. In order to search several treatment eﬀects including exploratory analyses, it is importantthat reducing of the calculation time is discussed. Kojima et al. (2013) derives the bartlett correction for
hypothesis testing in linear models with general error covariance matrices based on the general consistentestimators of the nuisance parameters. However, likelihood ratio(LR) test statistic need to be modified asa result of using the general consistent estimators, and the confidence interval of LR cannot be derived byexplicit. Furthermore, for LR and Score test statistic, REML function under null hypothesis is required, but
it was not proposed.
In our research, we show the bartlett correction for the multivariate random eﬀects meta-analysis model
via analytical approach and the explicit confidence interval of Wald, LR, Score test statistic based on theML and REML of nuisance parameters. Welham et al. (1997) shows the REML function under the null
hypothesis by using submodel, we propose novel REML function for the our research.

Reference[1] Gurevitch J., Koricheva J., et al. (2018). Meta-analysis and the science of research synthesis. Nature, 555,March, 175-182.
[2] Noma H., Nagashima K., et al. (2018). Bartlett‐type corrections and bootstrap adjustments of likelihood‐based inference methods for network meta‐analysis. Statistics in Medicine, 37, 1178-1190.
[3] Friede T, Röver C., et al. (2017). Meta-analysis of few small studies in orphan diseases. Res. Syn. Meth,
8, 79-91.
[4] Kojima M. and Kubokawa T. (2013). Bartlett-type adjustments for hypothesis testing in linear models
with general error covariance matrices. Journal of Multivariate Analysis, 122, 162-174.
[5] Welham S. J. and Thompson R. (1997). Likelihood Ratio Tests for Fixed Model Terms using ResidualMaximum Likelihood. J. R. Statist. Soc. B, 59, 3, 701-714.